Precalculus Examples

$2 \log_{6} (3) + 3 \log_{6} (8) - 5 \log_{6} (11)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{6} (3)$ by moving $2$ inside the logarithm.

$\log_{6} (3^{2}) + 3 \log_{6} (8) - 5 \log_{6} (11)$

Step 1.2

Raise $3$ to the power of $2$ .

$\log_{6} (9) + 3 \log_{6} (8) - 5 \log_{6} (11)$

Step 1.3

Simplify $3 \log_{6} (8)$ by moving $3$ inside the logarithm.

$\log_{6} (9) + \log_{6} (8^{3}) - 5 \log_{6} (11)$

Step 1.4

Raise $8$ to the power of $3$ .

$\log_{6} (9) + \log_{6} (512) - 5 \log_{6} (11)$

Step 1.5

Simplify $- 5 \log_{6} (11)$ by moving $5$ inside the logarithm.

$\log_{6} (9) + \log_{6} (512) - \log_{6} (11^{5})$

Step 1.6

Raise $11$ to the power of $5$ .

$\log_{6} (9) + \log_{6} (512) - \log_{6} (161051)$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{6} (9 \cdot 512) - \log_{6} (161051)$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{6} (\frac{9 \cdot 512}{161051})$

Step 4

Multiply $9$ by $512$ .

$\log_{6} (\frac{4608}{161051})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log_{6} (\frac{4608}{161051})$

Decimal Form:

$- 1.98348451 \dots$